"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."

We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they&apos;ve had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.

"... Introduction Adding constraints to a basic CSP model can signi cantly reduce search, e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4] and (less so) on generating impl ..."

Introduction Adding constraints to a basic CSP model can signi cantly reduce search, e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4] and (less so) on generating implied constraints [5]. We describe an approach to generating implied, symmetry breaking and specialisation constraints and apply this technique to quasigroup construction [10]. Given a problem class parameterised by size, we use a basic model to solve small instances with the Choco constraint programming language [7]. We then give these solutions to the HR automated theory formation program [1] which detects implied constraints (proved to follow from the speci cations) and induced constraints (true of a subset of solutions). Interpreting HR&apos;s results to reformulate the model can lead to a reduction in search on larger instances. It is often more ecient to run HR, interpret the results and so

"... We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were sup ..."

We report on the application of the HR program (Colton, Bundy, &amp; Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.

"... The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multi-agent version of HR which may ..."

The HR program, Colton et al. (1999), performs theory formation in domains of pure mathematics. Given only minimal information about a domain, it invents concepts, make conjectures, proves theorems and finds counterexamples to false conjectures. We present here a multi-agent version of HR which may provide a model for how individual mathematicians perform separate investigations but communicate their results to the mathematical community, learning from others as they do. We detail the exhaustive categorisation problem to which we have applied a multi-agent approach. 1

"... Automated Theorem Proving (ATP) researchers who always use the same problems for testing their systems, run the risk of producing systems that can solve only those problems, and are weak on new problems or applications. Furthermore, as the state-of-the-art in ATP progresses, existing test problems b ..."

Automated Theorem Proving (ATP) researchers who always use the same problems for testing their systems, run the risk of producing systems that can solve only those problems, and are weak on new problems or applications. Furthermore, as the state-of-the-art in ATP progresses, existing test problems become too easy, and testing on them provides little useful information. It is thus important to regularly nd new and harder problems for testing ATP systems. HR is a program that performs automated theory formation in mathematical domains, such as group theory, quasigroup theory, and ring theory. Given the axioms of the domain...

"... Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperat-ing reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on pro ..."

Using the achievements of my research group over the last 30+ years, I provide evidence to support the following hypothesis: By complementing each other, cooperat-ing reasoning process can achieve much more than they could if they only acted individually. Most of the work of my group has been on pro-cesses for mathematical reasoning and its applica-tions, e.g. to formal methods. The reasoning pro-cesses we have studied include: Proof Search: by meta-level inference, proof planning, abstraction, analogy, symmetry, and reasoning with diagrams. Representation Discovery, Formation and Evolution: by analysing, diagnosing and repairing failed proof and planning attempts, forming and repairing new concepts and conjectures, and forming logical representations of informally stated problems. ¤I would like to thank the many colleagues with whom I have worked over the last 30+ years on the research reported in this paper: